Is shakhkheer one of the characteristics of the letter kha?
The characteristics of the Arabic letters which are used in application while reading the Qur'an which are related to running of breath or running sound are:
Hams (running of breath) or Jahr (imprisonment of running of breath)
Rikhaawah (running of sound) or Shiddah (imprisonment of running of sound) or Tawassit (partial running and partial imprisonment of running of sound).
Shakheer means snoring. It is quite possible some Arabic language linguistic specialists listed this as a characteristic for the خاء but this is not something we apply or avoid while reading the Qur'an and not of the characteristics outlined by Imam Ibn Al Jazaree in his prose on tajweed rules.